Skinner-Rusk formalism for k-contact systems

TL;DR

This paper extends the Skinner-Rusk formalism to k-contact systems, unifying Lagrangian and Hamiltonian approaches for non-conservative field theories, especially useful for singular systems with dissipation.

Contribution

It introduces a Skinner-Rusk unified framework for k-contact systems, integrating Lagrangian and Hamiltonian formalisms into a single geometric setting.

Findings

Unified framework simplifies handling of singular systems.

Incorporates second-order conditions naturally.

Facilitates constraint algorithms and Legendre transformation.

Abstract

In previous papers, a geometric framework has been developed to describe non-conservative field theories as a kind of modified Lagrangian and Hamiltonian field theories. This approach is that of $k$-contact Hamiltonian systems, which is based on the $k$-symplectic formulation of field theories as well as on contact geometry. In this work we present the Skinner--Rusk unified setting for these kinds of theories, which encompasses both the Lagrangian and Hamiltonian formalisms into a single picture. This unified framework is specially useful when dealing with singular systems, since: (i) it incorporates in a natural way the second-order condition for the solutions of field equations, (ii) it allows to implement the Lagrangian and Hamiltonian constraint algorithms in a unique simple way, and (iii) it gives the Legendre transformation, so that the Lagrangian and the Hamiltonian formalisms are obtained straightforwardly. We apply this description to several interesting physical examples: the damped vibrating string, the telegrapher's equations, and Maxwell's equations with dissipation terms.